Tunning Pss Using GA

8/9/2019 Tunning Pss Using GA

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ELSEVIER

ELECTRIC

POUIER

SY rems

RESEIMH

Electric Power System s Research 39 (1996) 137 143

Tuning of power system stabilizers using genetic algorithms

Y.L. Abdel-Magid *, M.M. Dawoud

Electrica l Engineering Department. King Fahd Lnicersit~~ c~f Petroleum und Minerals, Dhahran 31261, Saudi Arabia

Received 21 June 1996: accepted 3 July 1996

Abstract

Several techniques exist for developing optimal controllers. This paper investigates the tuning of power system stabilizers (PSS)

using genetic algorithms (GA). A digital simulation of a linearized model of a single-machine infinite bus power system at some

operating point is used in conjunction with the genetic algorithm optimization process. The integral of the square of the error and

the time-multiplied absolute value of the error performance indices are considered in the search for the optimal PSS parameters.

In order to have good damping characteristics over a wide range of operating conditions, the PSS parameters are optimized

off-l ine for a selected set of grid points in the real power (P)-reactive power (Q) domain. The optimal settings thus obtained can

then be stored and retrieved on-line to update the PSS parameters based on measurements of the generator real and reactive

power. T ime domain simulations of the system with GA-tuned PSS show the improved dynamic performance under widely

varying load conditions. 0 1996 Elsevier Science S.A.

Keywords: Dynamic stability; Genetic algorithm: Power system stabilizer

1. Introduction

The application of genetic algorithms (GA) has re-

cently attracted the attention of researchers in the con-

trol area [l-4]. From the literature it is clearly seen that

genetic algorithms can provide powerful tools for opti-

mization. The present paper demonstrates the use of

GA to tune the parameters of a power system stabilizer

(PSS).

The use of high-speed excitation systems has long

been recognized as an effective method of increasing

stability limits. Static excita tion systems appear to offer

the practical ultimate in high-speed performance,

thereby providing a gain in stability limits. Unfortu-

nately. the high speeds and gains that give them this

capability also result in poor system damping under

certain conditions of loading [5]. To offset this effect

and to improve the system damping in general, supple-

mentary stabilizing signals are introduced in the excita-

tion systems through fixed parameters lead/lag power

system stabilizers [6,7]. The parameters of the PSS are

normally fixed at certain values which are determined

under a particular operating condition.

* Corresponding author. Tel.: + 966 3 8602277: fax: + 966 3

8603535.

037%7796:96 15.00 8 1996 Elsevler Scienc e S..4 Al l rights reserved.

PI1 SO378-7796(96)01 105-4

It is important to recognize that machine parameters

change with loading, making the dynamic behavior of

the machine quite different at different operating

points. Since these parameters change in a rather com-

plex manner [8,9], a set of PSS parameters which

provide good dynamic performance under a certain

operating condition may no longer yield satisfactory

results when there is a drastic change in the operating

point. Therefore, it is necessary to adapt the PSS

parameters in real time based on measurements of the

machine loading. In this paper, the optimum values of

the PSS parameters are determined off-line using a

genetic algorithm. The procedure is repeated for a

selected set of grid points in the real power/reactive

power domain. The optimum settings thus obtained can

be stored in a look-up table . Based on the on-line

measurement of values of P and Q, the PSS parameters

are updated in step with changes in the operating

conditions in the ensuing steady state [lo].

Scalar integral performance indices have proved to

be the most meaningful and convenient measures of

dynamic performance [ 11,121. Two performance indices

have been chosen for this study: they are the popular

integral of the squared error (BE) and the integral of

time-multiplied absolute value of the error (ITAE), and

are given, respectively, by


2/7

138

Y.L. Abdel-Magid, M.M. Dawoud : Electric Power Systems Research 39 (1996) 137- 143

s,= x

s

e(t) dr

(1)

0

s, =

s

x

+O)l dt (2)

0

The system to be studied is that of a single machine

connected to an infinite bus through a transmission

line. The PSS considered is a derivative-type power

stabilizer.

A digital simulation is used in conjunction with the

genetic algorithms optimization process to determine

the optimum parameters of the PSS for the perfor-

mance indices considered. Genetic algorithms are used

as parameter search techniques which utilize the genetic

operators to find near-optimal solutions. The advantage

of the GA technique is that it is independent of the

complexity of the performance index considered. It

suffices to specify the objective function and to place

finite bounds on the optimized parameters.

2. Genetic algorithms

Genetic algorithms (GA) are global search tech-

niques, based on the operations observed in natural

selection and genetics [l]. They operate on a population

of current approximations-the individuals-initially

drawn at random, from which improvement is sought.

Individuals are encoded as strings (chromosomes) con-

structed over some particular alphabet, e.g., the binary

alphabet {0, 1 , so that chromosome values are

uniquely mapped onto the decision variable domain.

Once the decision variable domain representation of the

current population is calculated, individual perfor-

mance is assumed according to the objective function

which characterizes the problem to be solved. It is also

possible to use the variable parameters directly to repre-

sent the chromosomes in the GA solution.

At the reproduction stage, a fitness value is derived

from the raw individual performance measure given by

the objective function, and used to bias the selection

process. Highly fit individuals will have increasing op-

portunities to pass on genetically important material to

successive generations. In this way, the genetic al-

gorithms search from many points in the search space

at once and yet continually narrow the focus of the

search to the areas of the observed best performance.

The selected individuals are then modified through

the application of genetic operators, in order to obtain

the next generation. Genetic operators manipulate the

characters (genes) that constitute the chromosomes di-

rectly, following the assumption that certain genes

code, on average, for fitter individuals than other genes.

Genetic operators can be divided into three main cate-

gories [2]: selection, crossover and mutation.

infinite bus

Fig. 1. Single machine connected to an infinite bus.

1. Selection: selects the fittest individuals in the current

population to be used in generating the next popula-

tion.

2. Cross-over: causes pairs, or larger groups of individ-

uals, to exchange genetic information with one an-

other.

3. Mutation: causes individual genetic representations

to be changed according to some probabilistic rule.

Genetic algorithms are more likely to converge to

global optima than conventional optimization tech-

niques, since they search from a population of points,

and are based on probabilistic transition rules. Conven-

tional optimization techniques are ordinarily based on

deterministic hill-climbing methods, which, by defini-

tion, will only find local optima. Genetic algorithms can

also tolerate discontinuities and noisy function evalua-

tions.

3. Problem formulation

The system considered in this paper is a synchronous

machine connected to an infinite bus through a trans-

mission line, as shown in Fig. 1. The synchronous

machine is described by a fourth-order model. The

relations in the block diagram shown in Fig. 2 apply to

a two-axis machine representation with a field circuit in

the direct axis but without damper windings. The inter-

action between the speed and voltage control equations

of the machine is expressed in terms of six constants

K,

-Kh.

These constants with the exception of

K3,

which

is only a function of the ratio of the impedance, are

dependent upon the actual real and reactive power

loading as well as the excitation levels in the machine

I I

Fig. 2. System block diagram


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Y.L. Abdel-Magid, M.M. Dawoud: Electru Power Systems Research 39 (1996) 137- 143

139

RANDOMLY GENERATE

NITAL. POPULATION

AND EVALUATE THE

PERFORMANCE INDEX

. SELECTlON

l CROSS-OVER

. MUTATION

GEN. =G&.+l

I

Fig. 3. Genetic algorithm flow chart

[8]. The equations describing the steady-state operation

of a synchronous generator connected to an infinite bus

through an external reactance can be linearized about

any particular operating point as follows:

APm-AP=iVl~

(3)

AP = K, Ab + K, AE:,

(4)

Fig. 4. The GA computed PSS parameter K for the entire loading

Fig. 6. The GA computed PSS parameter K for the entire loading

range (P, Q)= (0.1. . . 1.0; - 0.3, . . 1.0) (ISE).

range (P .Q) = (0.1, . . 1.0: - 0.3. . . 1.0) (ITAE).

Fig. 5. The GA computed PSS parameter T for the entire loading

range

(P. Q)=

(0.1, .__, 1.0; -0.3, . . 1.0) (ISE).

AE:, =

KS

AEM -

K&

1 + ST&K,

1+ ST&K,

A6

AV,=K,Ad+K,AE;

(6)

The constants K,-K6 are given in Appendix A.

The system data are as follows:

Machine (p.u.):

x,,= 1.6; x; = 0.32;

xy = 1.55

L, = 1 o:

w,,= 1207~ rad ss; TX = 6.0 s

D = 0.0; M= 10.0

Transmission line (p.u.):

r, = 0;

x, = 0.4

Exciter:

K, = 50.0;

T, = 0.05 s

Loading (p.u.)

P = (0.1, 0.2 )..., 1.0);

Q = ( - 0.3, - 0.2 ,..., 1.0)

The supplementary stabilizing signal considered in

this paper is one proportional to electrical power. A

derivative-type power stabilizer is used. Its transfer

function is given by:

G,(s) =

KS

(s + l/7J2

30

1


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140 Y.L. Ahdel-Magid. M.M. Dawoud- Electrw Pmwr System Research 39 (1996) 137- 143

0.8

0.6

+ 0.4

0.2

0

1

Fig. 7. The GA computed PSS parameter T for the entire loading

range (P, Q) = (0.1. .._, 1.0: -0.3. ,,., 1.0) (ITAE ).

where K and Tare the PSS parameters to be optimized.

This particular configuration of the PSS was chosen as

the result of extensive studies of root-locus diagrams

for numerous system loading conditions [lo] .

In this paper, the optimum values of the parameters

K and T, which minimize the different performance

indices selected, are easily and accurately computed

using genetic algorithms. For a given operating point,

the equations describing the machine, excitation con-

trol, transmission line and the PSS, are simulated and

the performance index evaluated. In a typical run of the

GA, an init ial population is randomly generated. This

init ial population is referred to as the 0th generation.

Each individua l in the init ial population has an associ-

ated performance index value. Using the performance

index information, the GA then produces a new popu-

lation. The application of a genetic algorithm involves

repet itively performing two steps:

1. The calculation of the performance index for each of

the individuals in the current population. To do this,

the system must be simulated to obta in the value of

the performance index.

2. The genetic algorithm then produces the next gener-

ation of individuals using the selection, crossover

and mutation operators.

These two steps are repeated from generation to gener-

ation until the population has converged. producing the

optimum parameters. The procedure is then repeated

for the selected set of grid points in the real poweqreac-

tive power domain (P=O.l, 0.2, . . . 1.0; Q= -0.3.

- 0.2, . ., 1.0). A flow-chart of the genetic algorithm

optimization procedure is given in Fig. 3.

Table 1

Variation of T with P and Q (ISE)

P=O.l P = 0.4 P=O.7

P= 1.0

Q = -0.3 1.56

0.38 0.26 0.27

Q = 0.0 1.14 0.61 0.38 0.3

Q = 0.3 0.89 0.7 0.48 0.4

Q = 0.7 0.75

0.68 0.58 0.49

Q= 1.0 0.76 0.7 0.63 0.54

Table 2

Variation of K with P and Q (ISE)

P=O.l P = 0.4

P = 0.7 P= 1.0

Q = -0.3

1.69 I .03 2.26 3.07

Q = 0.0 5.14 1.68

1.93 2.66

Q = 0.3 9.50 2.74 2.3

2.56

Q = 0.7 15.32 4.26 3 2.84

Q= 1.0 18.99 5.19

3.47 3.08

Penalizing only speed excursions, the two perfor-

mance indices considered in this study take the form:

s

-1

ISE = (Au) dt

(8)

0

ITAE =

i

-J tlA& dt

(9)

0

To compute the optimum parameter values, a unit step

disturbance in mechanical power was used to perturb

the system from its operating point.

4. Results and tests

The GA computed PSS parameters are plotted in

Figs. 4-7 in three-dimensional form for the entire

load ing range considered. while Tables l-4 list the

values of these parameters for some selected loading

conditions. Tables 1 and 2 show the optimum values of

K and T when the performance index is the ISE. The

optimum values of K and T for the ISE case are shown

in Figs. 4 and 5 for the entire loading range.

Tables 3 and 4 show the optimum values of K and T

when the performance index is the ITAE. The optimum

values of K and T for the ITAE case are shown in Figs.

6 and 7 for the entire loading range.

It is clear that the optimal settings of the PSS

parameters vary widely with the operating poin t and

also depend on the performance index being minimized.

Simulation tests for various operat ing points and with

the GA computed optimum parameter settings were

performed. Figs. 8- 13 are sample simulation results

illustrating some significant aspects of the system re-

sponse. A step-like disturbance in mechanical power

was used to perturb the system from its operating

Table 3

Variation of ;r with P and Q (ITAE)

Q= -0.3

Q = 0.0

Q = 0.3

Q = 0.7

Q= 1.0

P=O.l P = 0.4

P = 0.7 P= 1.0

0.74 0.21 0.18 0.20

0.58 0.34

0.24 0.23

0.47 0.38 0.29 0.26

0.42 0.38 0.34 0.31

0.41 0.39

0.37 0.36


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Y.L.. Abdel-Magid, M.M. Duwoud Elrcrric Power S~.stem.s Reseurch 39 (1996) 137- 143 141

Table 4

Variation of K with P and Q (ITAE)

P = 0.1 P = 0.4

P=O.7 P= 1.0

Q =

-0.3 2.01

1.58

2.54 3.13

Q = 0.0 5.87 1.85 2.18 2.75

Q =

0.3

10.8

2.97 2.48 2.69

Q=O.7 16.94 4.55 3.19 2.98

Q= 1.0 20.99 5.55 3.68 3.2

point. The traces shown are for speed deviations and

they are for three cases:

1. PSS disconnected.

2. PSS parameters are those resulting from minim izing

the ISE.

3. PSS parameters are those resulting from minim izing

the ITAE.

From the results, it is observed that:

1. The PSS considered significantly increases the stabil-

ity of the system.

2. Altering the PSS parameters to cope with realjreac-

tive power loading maintains optimum performance

for a wide range of system operating conditions.

3. The ITAE is superior to the ISE as far as the

damping of the oscillations and the settling time are

considered.

It is worth noting that the results obtained for the

ISE case agree with those available in the literature [lo],

where Lyapunov functions and more elaborate classical

optimization techniques were used. The results obtained

for the ITAE have not been previously obtained due to

the complexity of problem formulation. Because of the

simplicity of using the GA with complex objective

functions and nonl inear systems, other performance

indices and ful l system representations are being cur-

rently investigated.

: ..

.

-. :

-*:

-0.03

time, s

0

2 4 6 8 10

Fig. 8. Spee d deviation for (P. Q) = (0.1. - 0.3): no stab ilizer (. ):

Fig. 10 . Spee d deviation for (P. Q) = (0.1, 1.0): no stabiliz er (. .);

ISE (p -); ITA E (--).

1SE (m ): ITAE(----).

0.02

A%.of5.4

;T

: . .:

. . >

0.01

0.005

0.

-0.005.

-0.01 -

-0.015

-

: :

;:

:;

i

time,s 1

-0.02L

0 2

4

6 8 10

Fig. 9. Spee d deviation for (P, Q) = (0.4, 0.3): no stabiliz er (. ):

ISE (- I: ITAE (p,.

5. Conclusions

The tuning of power system stabilizers (PSS) using

genetic algorithms (GA) is investigated in this paper. A

digital simulation of a linearized model of a single-ma-

chine infinite bus power system at some operating point

is used in conjunction with the genetic algorithm opti-

mization process.The integral of the square of the error

and the time-multiplied absolute value of the error

performance indices are considered. For each perfor-

mance index, the PSS parameters are optimized off-line

for a selected set of grid points in the real power-reac-

tive power (P-Q) domain. It is suggested hat the PSS

parameters be updated based on measurements of the

generator real and reactive power. Time domain simu-

lations of the system with GA-tuned PSS show the

improved dynamic performance under widely varying

load conditions and points to the superiority of the

ITAE performance index relative to the ISE.

time, s

1

-0.021

0 2

4 6 8 10


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142

Y.L. Abdel-Magid, M.M. Dawoud /Electric Power Systems Research 39 (1996) 137-143

0.06l

tiwfe,s

0 2 4 6 8

10

Fig. 1 I. Speed deviation for (P. Q) = (1.0, -0.3): no stabilizer

(. .); ISE (- - -): ITAE (---).

Acknowledgements

The authors acknowledge the support and encour-

agement of King Fahd University of Petroleum and

Minerals which made it possible to conduct this re-

search.

Appendix A

All variables with subscript 0 are values of the vari-

ables evaluated at their pre-disturbance steady-state

operat ing poin t from known values of VtO, P, and Q, as

given by Eqs. (lo)-( 16). All variables preceded by A are

deviations of these variables from their values at the

steady-state operat ing point. The constants K,-K6 are

given in Eqs. (17)-(22).

POvto

(10)

(11)

vqo = JV - t&

(12)

0.02

A0

: .* . .

-0.015. ;:

time,s

0 2

4 8 8 10

Fig. 12. Spee d deviation for (P, Q) = (0.7, 0.0): no stabiliz er ( ):

ISE (p -); ITA E (---).

Fig. 13. Spee d devia tion for (P, Q) = (1 .O, 1.0): no stabiliz er (. );

ISE (- - m); ITA E (-).

i Q. + xyi&Q. + xyiio

V Y OYO

E,, = uqo + &xy,, = uqo + &xy

E. = J( vdo + x,i,,) + ( vf l - x&J2. = \i( vdo + x,z,,) + ( vf l - x&J2

6, = tan - , = tan -

(%I + &dqo)%I + x&o)

(c,o - x,&Jc,o - x,&J

K, =, =

x,-xl.,-xl.

E,,E, cos 6,,,E, cos 6,

-Y, + x:,Y, + x:,

lNEo sin 6, +NEo sin 6, +

*L + Xl,L + Xl,

K

2

= E, sin S,E, sin S,

x, + x : ,, + x : ,

xi + x,i + x,

KS = ~S = ~

xc/ + xec/ + xe

K4 = ___

x,1 xl/ E sin ho

x,+x;

K5L3 --

&

vcmE, cos 6, - - -

Vyo

x, + .q v,,

x,+x& VT0

&=- L-

El/o

-Yc+ x& v,,

E, sin 6,

(13)

(14)

(15)

(16)

(17)

(18)

(19)

W)

(21)

G-9)

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[II

VI

VI

[41

[51

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Tunning Pss Using GA

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